二维功能梯度壁板热颤振本征问题的精确解

为了得到二维功能梯度壁板热颤振的精确解并揭示颤振机理，根据经典薄板理论及一阶活塞理论，建立了超声速气流下二维功能梯度壁板的本征控制微分方程并求得了精确解，根据得到的本征根对颤振机理进行了分析。针对功能梯度材料（FGM）的不同体积分数，分别研究了壁板在恒温场及非线性温度场下的颤振边界随马赫数的变化规律，并比较了2种温度场下的结果。通过分析简支、固支及其组合边界情况下的壁板颤振特性，从数学角度发现颤振现象的发生是由于挠度的一阶导数导致刚度非对称，且功能梯度材料能够有效提高热环境下壁板的颤振边界，同时利用ABAQUS软件对功能梯度壁板的振动特性进行了模拟，进一步验证了所提方法的有效性。 

Exact solutions of thermal flutter of two-dimensional functionally graded panel

For achieving the thermal flutter exact solutions of two-dimensional functionally graded panel and revealing the mechanism of the thermal flutter, based on the classical thin plate theory and the first-order piston theory, the characteristic governing differential equation of two-dimensional functionally graded panel in supersonic flow is established and exact solutions are obtained. Through the analysis of the eigenvalues, the mechanism of the panel flutter is investigated. According to different volume fraction of Functionally Graded Materials (FGM), the flutter boundary changes with Mach number in constant temperature field and nonlinear temperature field are studied respectively, and the results in two temperature fields are compared. By analyzing the flutter characteristics of panels with simply supported, fixed and integrated edges, it can be concluded that the flutter phenomenon is caused by the first-order derivative of deflection which leads to the asymmetry of system stiffness, and FGM can effectively improve the flutter boundary of the panel in the thermal environment. Meanwhile, the vibration properties of the FGM panel is simulated with ABAQUS, further validating the effectiveness of the present method.